Skip to main content
SpringerLink
Log in

Estimating the number of basins of attraction of multi-objective combinatorial problems

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

The efficiency of local search is proportional to the number and the distribution of basins of attraction. Often combinatorial optimisation problems have a large number of local optima, uncountable with available computational resources. Approximating the number of basins of attraction and the minimal number of samples for visiting all basins at least once are complex problems for which we assume specific distributions of basins of attraction. We define two types of basins of attraction of multi-objective combinatorial optimisation problems with complementary properties. Acknowledging that each local search run generates a Pareto front of solutions, either each Pareto local solution corresponds to a basin of attraction, or a Pareto basin matches an entire Pareto local front. Simulations compare parametric and non-parametric estimators on bi-objective quadratic assignment problem instances.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Alyahya K, Rowe JE (2016) Simple random sampling estimation of the number of local optima. In: International conference on parallel problem solving from nature (PPSN). Springer, New York, pp 932–941

  • Bilodeau M, Brenner D (2008) Theory of multivariate statistics. Springer, New York

    MATH  Google Scholar 

  • Burnham KP, Overton WS (1978) Estimation of the size of a closed population when capture probabilities vary among animals. Biometrika 65(3):625–633

    Article  MATH  Google Scholar 

  • Buzas J, Dinitz J (2014) An analysis of NK landscapes: interaction structure, statistical properties, and expected number of local optima. IEEE Trans Evolut Comput 18(6):807–818

    Article  Google Scholar 

  • Drezner Z, Hahn PM, Taillard ÉD (2005) Recent advances for the quadratic assignment problem with special emphasis on instances that are difficult for meta-heuristic methods. Ann Oper Res 139(1):65–94

    Article  MathSciNet  MATH  Google Scholar 

  • Durrett R (2013) Probability: theory and examples. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Ehrgott M (2005) Multicriteria optimization. Springer, New York

    MATH  Google Scholar 

  • Elorza A, Hernando L, Mendiburu A, Lozano JA (2018) Estimating attraction basin sizes of combinatorial optimization problems. In: Progress in artificial intelligence

  • Eremeev A, Reeves CR (2002) Non-parametric estimation of properties of combinatorial landscapes. In: Applications of evolutionary computing. Springer, Berlin, pp 31–40

  • Feller W (1950) An introduction to probability theory and its applications. Wiley, Hoboken

    MATH  Google Scholar 

  • Garnier J, Kallel L (2001) Efficiency of local search with multiple local optima. SIAM J Discrete Math 15(1):122–141

    Article  MathSciNet  MATH  Google Scholar 

  • Goldman BW, Punch WF (2016) Hyperplane elimination for quickly enumerating local optima. In: Evolutionary computation in combinatorial optimization EvoCOP, pp 154–169

  • Happel R, Stadler PF (1996) Canonical approximation of fitness landscapes. Complexity 2(1):53–58

    Article  Google Scholar 

  • Hernando L, Mendiburu A, Lozano JA (2013) An evaluation of methods for estimating the number of local optima in combinatorial optimization problems. Evolut Comput 21(4):625–658

    Article  Google Scholar 

  • Knowles JD, Corne DW (2000) Approximating the nondominated front using the Pareto archived evolution strategy. Evolut Comput 8(2):149–172

    Article  Google Scholar 

  • Kollo T, von Rosen D (2005) Advanced multivariate statistics with matrices. Springer, New York

    Book  MATH  Google Scholar 

  • Liu C, Rubin DB (1995) ML estimation of the multivariate t-distribution with unknown degrees of freedom. Stat Sin 5:19–39

    Google Scholar 

  • Ochoa G, Tomassini M, Vérel S, Darabos C (2008) A study of NK landscapes’ basins and local optima networks. In: Proceedings of the conference on genetic and evolutionary computation (GECCO), ACM, pp 555–562

  • Paquete L, Stützle T (2006) A study of stochastic local search algorithms for the biobjective QAP with correlated flow matrices. Eur J Oper Res 169(3):943–959

    Article  MathSciNet  MATH  Google Scholar 

  • Prügel-Bennett A, Tayarani-Najaran M-H (2012) Maximum satisfiability: anatomy of the fitness landscape for a hard combinatorial optimization problem. IEEE Trans Evolut Comput 16(3):319–338

    Article  Google Scholar 

  • Reeves CR, Eremeev AV (2004) Statistical analysis of local search landscapes. JORS 55(7):687–693

    Article  MATH  Google Scholar 

  • Reidys CM, Stadler PF (2002) Combinatorial landscapes. SIAM Rev 44(1):3–54

    Article  MathSciNet  MATH  Google Scholar 

  • Srinivas N, Deb K (1994) Multiobjective optimization using nondominated sorting in genetic algorithms. Evolut Comput 2(3):221–248

    Article  Google Scholar 

  • Verel S, Ochoa G, Tomassini M (2011) Local optima networks of NK landscapes with neutrality. IEEE Trans Evolut Comput 15(6):783–797

    Article  Google Scholar 

  • Wishart J (1928) The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A(1–2):32–52

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ITLearns.Online, Utrecht, The Netherlands

    Madalina M. Drugan

Authors
  1. Madalina M. Drugan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Madalina M. Drugan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Drugan, M.M. Estimating the number of basins of attraction of multi-objective combinatorial problems. J Comb Optim 37, 1367–1407 (2019). https://doi.org/10.1007/s10878-018-0357-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-018-0357-8

Keywords

Search

Navigation